Bifurcations in Boolean Networks

descriptionPublicationkeyboard_double_arrow_right Article , Preprint , Conference object 01 Jan 2011Embargo end date: 01 Aug 2011 English Publisher:Centre pour la Communication Scientifique Directe (CCSD)Journal:Discrete Mathematics & Theoretical Computer Science, volume DMTCS Proceedings vol. AP,... (eissn: 1365-8050,

Copyright policy )Funded by:NSF | Tracking the West Antarct..., NSF | Collaborative Research: N..., NSF | Collaborative Research: M... +6 projects

Authors: Chris Kuhlman; Henning Mortveit; David Murrugarra; Anil Kumar;

doi: 10.46298/dmtcs.2975 , 10.48550/arxiv.1108.2974

arXiv: 1108.2974

Bifurcations in Boolean Networks

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Abstract

This paper characterizes the attractor structure of synchronous and asynchronous Boolean networks induced by bi-threshold functions. Bi-threshold functions are generalizations of standard threshold functions and have separate threshold values for the transitions $0 \rightarrow $1 (up-threshold) and $1 \rightarrow 0$ (down-threshold). We show that synchronous bi-threshold systems may, just like standard threshold systems, only have fixed points and 2-cycles as attractors. Asynchronous bi-threshold systems (fixed permutation update sequence), on the other hand, undergo a bifurcation. When the difference $\Delta$ of the down- and up-threshold is less than 2 they only have fixed points as limit sets. However, for $\Delta \geq 2$ they may have long periodic orbits. The limiting case of $\Delta = 2$ is identified using a potential function argument. Finally, we present a series of results on the dynamics of bi-threshold systems for families of graphs.

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Keywords

bi-threshold, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [info.info-dm] computer science [cs]/discrete mathematics [cs.dm], [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], [NLIN.NLIN-CG] Nonlinear Sciences [physics]/Cellular Automata and Lattice Gases [nlin.CG], Dynamical Systems (math.DS), [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], Boolean networks, sequential dynamical systems, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], QA1-939, threshold, FOS: Mathematics, [NLIN.NLIN-CG]Nonlinear Sciences [physics]/Cellular Automata and Lattice Gases [nlin.CG], Mathematics - Dynamical Systems, [math.math-ds] mathematics [math]/dynamical systems [math.ds], [math.math-co] mathematics [math]/combinatorics [math.co], asynchronous, boolean networks, synchronous, graph dynamical systems, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], bifurcation, Mathematics, [nlin.nlin-cg] nonlinear sciences [physics]/cellular automata and lattice gases [nlin.cg]

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